Discussion
The correct ordering of the growth rates \(3^{n}\), \(n^{3}\), \(3^{3}\), \(\log_{3} n\) is :
Answer: D
The correct order is \(3^{3},\ \log_{3} n,\ n^{3},\ 3^{n}\).
\(3^{3} = 27\) is a constant, which does not grow at all. A logarithm grows, but more slowly than any power of n. A polynomial \(n^{3}\) grows more slowly than any exponential function, so \(3^{n}\) is the fastest:
\[ O(1) \lt O(\log n) \lt O(n^{3}) \lt O(3^{n}) \]
Options (A) and (B) put \(\log_{3} n\) below a constant, which is wrong for large n, and (A) also puts the constant after \(n^{3}\). Option (C) puts \(3^{n}\) before \(n^{3}\), but the exponential eventually outgrows any polynomial.
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